Kirillov model
In mathematics, the Kirillov model, studied by Kirillov (1963), is a realization of a representation of GL2 over a local field on a space of functions on the local field.
If G is the algebraic group GL2 and F is a non-Archimedean local field, and τ is a fixed nontrivial character of the additive group of F and π is an irreducible representation of G(F), then the Kirillov model for π is a representation π on a space of locally constant functions f on F* with compact support in F such that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \pi({ab\choose 01})f(x) = \tau(bx)f(ax). }
Jacquet & Langlands (1970) showed that an irreducible representation of dimension greater than 1 has an essentially unique Kirillov model. Over a local field, the space of functions with compact support in F'* has codimension 0, 1, or 2 in the Kirillov model, depending on whether the irreducible representation is cuspidal, special, or principal.
The Kirillov model can also be defined for the general linear group GLn using the mirabolic subgroup.
The Whittaker model can be constructed from the Kirillov model, by defining the image Wξ of a vector ξ of the Kirillov model by
- Wξ(g) = π(g)ξ(1)
where π(g) is the image of g in the Kirillov model.
References
- Bernstein, Joseph N. (1984), "P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case)", Lie group representations, II (College Park, Md., 1982/1983), Lecture Notes in Math., vol. 1041, Berlin, New York: Springer-Verlag, pp. 50–102, doi:10.1007/BFb0073145, MR748505
- Kirillov, A. A. (1963), "Infinite-dimensional unitary representations of a second-order matrix group with elements in a locally compact field", Doklady Akademii Nauk SSSR, 150: 740–743, ISSN 0002-3264, MR0151552
- Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, MR0401654